Annual Report 2000 - WIT

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131
the both cases, quasi-harmonic and quasi-step-function medium excitation. Finally we
demonstrate the SBRC method on some case studies.
THE CONCEPT OF TRIGGERING FRONTS
In the following we approximate a real configuration of a fluid injection in a borehole
by a point source of pore pressure perturbation in an infinite heterogeneous anisotropic
poroelastic fluid-saturated medium. In the low-frequency limit of the Biot equations
(Biot 1962) the pore-pressure perturbation can be approximately described by the
following differential equation of diffusion:
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ÄÜ´• (1)
»P×]Ú
where »P×]Ú are components of the tensor of the hydraulic diffusivity, ÊÛÚ (ݽ‘¿Ì•?“Û•ÆÞ ) are
the components of the radius vector from the injection point to an observation point
in the medium and ˆ is the time. This equation corresponds to the second-type Biot
waves (the slow P-waves) in the low frequency limit and describes linear relaxation
of pore-pressure perturbations. Note, that this equation is valid for a heterogeneous
medium in respect of its hydraulic properties. In other words, components of the
tensor of the hydraulic diffusivity can be heterogeneously distributed in the medium.
The tensor of hydraulic diffusivity is directly proportional to the tensor of permeability
(see e.g. Rindschwenter et al. within this volume).
Ê1×ÙØ
ÊÛÚ
In some situations (e.g., some hydrofracturing experiments) the hydraulic diffusivity
can be changed considerably by the fluid injection. This means, that in the equation
above the diffusivity tensor must become pore-pressure dependent. Therefore, this
equation becomes non-linear. Such changes of the diffusivity take place in restricted
regions around boreholes. However, our method is aimed at estimating the effective
hydraulic diffusivity in a large rock volume of the spatial scale of the order of 1km.
Moreover, in a given elementary volume of the medium, the triggering of the earliest
microseismic events starts before the substantial relaxation of the pore-pressure
occurs. This means, that even in the 'near zone' very early events occur in the
practically unchanged medium. In other words, the front of significant changes of the
medium propagates behind the quicker triggering front of earlier microseismic events.
However, it is precisely these early events that are important for our approach for
estimating the diffusivity. Thus, the corresponding estimate should be approximately
equal to the diffusivity of the unchanged medium even in such situations, where the
diffusivity was strongly enhanced by the hydraulic fracturing. Because of this reason
we assume that changes of the diffusivity caused by the injection can be neglected.